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Model Order Reduction: Theory, Research Aspects and Applications

We have organized the book according to these goals. In Part I, the rationale behind model order reduction is explained, and an overview of the most common methods is described. Added to basket. Nonlinear Ordinary Differential Equations. Dominic Jordan. Schaum's Outline of Vector Analysis, 2ed. Murray R.

A Student's Guide to Fourier Transforms. The Calculus Lifesaver.


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Adrian Banner. How to Think About Analysis. Lara Alcock. Calculus Late Transcendentals. Howard Anton. Patrick Jones. Schaum's Outline of Calculus. Frank Ayres. An Introduction to Manifolds.

Data-Driven Control: The Goal of Balanced Model Reduction

Loring W. Robert C. The Cartoon Guide to Calculus. In the following sections we will present the methodology to select the linearization points and also the weighting procedure for the combination of the linearization points. Training trajectories form an integral part of the TPWL method which theoretically, should be able to cover all the domain of the nonlinear function.

The selection of training trajectories, therefore, requires careful selection of initial conditions upon which the trajectories of the swing model depend. One may consider it is inefficient to compute so many nonlinear functions to cover the whole domain. However, in practice there are only a few possible conditions a system can achieve in real time applications.

It is to be stressed that the TPWL method can interpolate between the training trajectories but not to extrapolate. Therefore, it is necessary to include all the trajectories in the training set that are considered to be visited by the nonlinear function [ 30 ]. Selection of linearization points has been done during the offline stage of the TPWL method and is performed on the full original dimension of the system.

This is the first step in the TPWL simulation after the full nonlinear solutions have been obtained at the predefined training trajectories. The linearized function approximates the actual nonlinear function using the Jacobian at the specified linearization points, represented by j in the superscript. The idea behind the TPWL method is that a number of linearization points are selected and the linearized functions at those snapshots are summed up by a weighting function given a global approximation as presented in Eq.

A detail on the selection of linearization points has been presented in Algorithm 1. The strategy to select the linearization points is traditionally based on the difference between the phase differences between the successive time steps, i. If the phase difference is greater than some angle, e. This method is widely used in the studies referenced in our work as presented in the works of Albunni [ 30 ] and Rewienski et al. The limitation in this method is that when the selection is based on the distance between points, there is no control over the error.

This can lead to significant error especially if the Jacobian is close to singular. The alternative method adopted in this study is to place the linearization points adaptively based on an error indicator. This is obtained by the difference between the nonlinear function and the approximation obtained by TPWL method.

Model Reduction of Dynamical Systems

When a point is encountered for which the error is greater than a given threshold, the TPWL model is enriched with a new point. This procedure is applied recursively to all the snapshots of the training simulations, starting with a single point that is the initial condition. A simple nonlinear trigonometric function and its linear approximation based on the concept of fixed distance in the context of its norm is plotted in Fig. As it is observable from Fig. Generally, the increase in number of linearization points is of no significant loss in computation time as the selection is done during the offline phase while in the online phase the computation of linear functions is very quick.

This approach is similar to the method proposed in the study of Liu et al.

The modified version of the selection of linearization points is more time consuming then the original method proposed by Albunni [ 30 ]. However, selection of linearization points is performed during the offline phase where time is not a constraint, the proposed method in the current study has higher accuracy with the problem discussed here. TPWL method combines the linearized model in a convex combination approximating the original nonlinear system.

In a convex combination all the coefficients are greater or equal to zero with the sum of all the coefficients equal to one. The calculation of weights is carried out during both the online and offline phases in the current study. A smaller value smooths the function and make its appearance continuous, while a higher value results in kinks in the function. Once, the weights are calculated with respect to all the points in the set S , the weights are normalized as:.

Once a linear model has been obtained from the training trajectories during the offline phase, the values of the nonlinear function and the Jacobian matrix evaluated at the selected linearization points are reduced and stored to be used during the online phase along with the set of linearization points. The nonlinear function and the Jacobian is projected in the reduced space as:. Since, we replace the nonlinear function with an approximation containing sum of linearized functions, we have. With the above information, we now have a system in reduced basis which fully exploits the benefits of reduced order modeling.

order reduction theory research Manual

As it can be observed from Eq. The mathematical model describing the ring grid is the swing dynamics given by the Eq. The grid in this study is a ring grid containing all the generator nodes and a slack node in a topology such that the slack node is connected with all the generators. Here, the slack bus is modeled as an infinite bus which is a simplifying assumption that the voltage at this bus is always constant and it has infinite power capacity as the impedance is zero for this bus. A list of assumptions for the grid in the current study are:.

A ring grid power network. Blue circles representing generators connected to an infinite bus. The generators are small and the ratio between the length of transmission line joining generators to the infinite bus and the length of transmission line joining two consecutive generators is much bigger. Hence, the interaction between a generator and infinite bus is much smaller than the interaction between two neighboring generators.

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Transmission lines joining two consecutive generators is shorter than the line joining the generators with the infinite bus. The Eq. The grid studied consists of generators and one slack node connected to all the generators. The data used in the study is given in Table 1 , all the values are given in per-unit system.

The values used in the current study are adapted from the study of Susuki et al. Also, the number of generators in the study of Susuki et al. Although different from the study by Susuki et al. In the current study, there are three different scenarios of initial conditions that must be taken into account for the training trajectories, these are listed in Table 3.

The dependence of the trajectories on the initial conditions is evident since the trajectory will be different in each case. These training trajectories were categorized into three different types.